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The Scalar, Vector and Tensor Contribution of a
Stochastic Background of Primordial Magnetic
Fields to CMB Anisotropies
Daniela Paoletti University and INFN of
Ferrara INAF/IASF-Bologna
Work in collaboration with Fabio Finelli and
Francesco Paci For more details The Impact of a
Stochastic Background of Primordial Magnetic
Fields on the Scalar Contribution to Cosmic
Microwave Background Anisotropies Finelli, Paci,
Paoletti Phys. Rev. D78 (2008) 023510 The Full
Contribution of a Stochastic Background of
Magnetic Fields to CMB Anisotropies Paoletti,
Finelli, Paci ArXiv0811.0230 to appear in
MNRAS
27th May 2009, School of Astrophysics, Bertinoro
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PRIMORDIAL MAGNETIC FIELDS
Magnetic fields are observed everywhere in the
universe from galaxies to galaxy clusters. These
observational evidences led to the hypothesis of
the existence of primordial magnetic fields (PMF)
generated in the early universe. If such fields
exist they may have left an imprint on cosmic
microwave background (CMB) anisotropies in
temperature and polarization. With the present
CMB data and the ones coming very soon.. is
therefore possible to investigate PMF and
constrain the parameters which characterize them.
There have been several studies on the effects of
PMF on CMB anisotropies (Giovannini et al.,
Kahniashvili and Ratra for scalar contribution,
Subramanian, Lewis, Durrer et al. for vector and
tensor respectively) our work improves on PMF
EMT Fourier power spectra and on the initial
conditions.
The simplest model of PMF supported in a
Robertson Walker universe is a stochastic
background of primordial magnetic fields (SB of
PMF).
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STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC
FIELDS
A SB of PMF does not carry neither energy density
nor pressure at the homogeneous level. The
absence of a background is the reason why even if
PMF are a relativistic massless and with
anisotropic stress component, like neutrinos(we
have considered only massless neutrinos) and
radiation, their behaviour is completely
different.
In the infinite conductivity limit (electric
fields nullify) PMF EMT becomes
And the magnetic fields and energy density simply
scales as
Conservation equations for PMF simply reduce to a
relation between PMF anisotropic stress, energy
density and the Lorentz force
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THE SCALAR, VECTOR AND TENSOR CONTRIBUTIONS OF A
STOCHASTIC BACKGROUND OF PRIMORDIAL MAGNETIC
FIELDS ON CMB
PMF induce three types of perturbations SCALAR,
VECTOR and TENSOR perturbations. They act on
primordial perturbations through three different
effects
PMF gravitate
Influence metric perturbations
PMF anisotropic stress
Adds to photon and neutrino ones
Lorentz force on baryons
Affects baryon velocity
Prior to the decoupling baryons and photons are
coupled by the Compton scattering
Lorentz force acts indirectly also on photons
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SCALAR CONTRIBUTION
PMF EMT acts as a source term in the Einstein
equations for metric perturbations
PMF induce a Lorentz force on baryons, the
charged particles of the plasma.
Where the conservation equations for baryons with
an electromagnetic source become
Primordial plasma is globally neutral
Energy conservation is not affected
Baryon Euler equation
During the tight coupling regime the photon
velocity equation is
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INITIAL CONDITIONS FOR SCALAR COSMOLOGICAL
PERTURBATIONS
We calculated the correct initial conditions
(Paoletti et al. 2008) truncating the neutrino
hierarchy at F40 instead of F30 as in our
previous work (Finelli et al. 2008).
The magnetic contribution drops from the metric
perturbations at leading order .This is due to a
compensation which nullifies the sum of the
leading contribution in the energy density in the
Einstein equations and therefore in metric
perturbations. There are similar compensations
also for a network of topological defects, which
does not carry a background EMT as this kind of
PMF.
C1 characterize the standard adiabatic mode
Paoletti et al. 2008 ArXiv0811.0230
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SCALAR FULLY MAGNETIC MODE
Note that the presence of PMF induces the
creation of a fully magnetic mode in metric and
matter perturbations. (This mode is the leading
one in radiation era for matter
perturbations.) This new indipendent mode is the
particular solution of the inhomogeneous Einstein
equations,where the homogeneous solution is
simply the standard adiabatic mode (or any other
isocurvature mode). This mode can be correlated
or uncorrelated with the adiabatic one like
happens for isocurvature modes, depending on the
physics which has generated the PMF. However, the
nature of the fully magnetic mode is completely
different from isocurvature perturbations and so
are its effects. The fully magnetic mode is the
particular solution of the inhomogeneous Einstein
system sourced by a fully inhomogeneous
component, while isocurvature modes are solution
of the homogeneous one where all the species
carry both background and perturbations.
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MAGNETICALLY DRIVEN VECTOR MODE
Vector perturbations are induced by vorticity in
the primordial plasma. Even if a primordial
vorticity is considered in RW it decays rapidly
and primordial vector mode as a consequence
rapidly disappears. Vector pertubations can be
sourced by a dishomogeneous SB of PMF.
Vector perturbations have vanishing energy
density vector metric perturbations are sourced
by the anisotropic stress in the plasma. Carrying
anisotropic stress PMF source vector
perturbation.
As for the scalar mode, also in the vector case
is necessary to take into account the Lorentz
force induced on baryons. Therefore PMF also
modify the vector part of the baryon velocity.

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TENSOR CONTRIBUTION
The tensor primordial perturbations, namely
primordial gravitational waves, represent one of
the key predictions of the standard inflationary
model. PMF carrying anisotropic stress generate
an independent mode in addition to the
inflationary one.
Tensor metric pertubation are sourced by the
anisotropic stress in the plasma.
The tensor initial conditions in the presence of
PMF are
Paoletti et al. 2008
PMF are responsable for the new leading term in
the neutrino anisotropic stress otherwise absent.
This is the so-called compensation between
collisionless components and PMF which strongly
modifies the effect of PMF on tensor modes
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MAGNETIC FIELDS POWER SPECTRUM
We considered a power law power spectrum PMF
In order to consider the damping of PMF on small
scales due to radiation viscosity we considered a
sharp cut off in the power spectrum at a scale
kD.With this cut off the two point correlation
function of PMF is
PMF EMT POWER SPECTRUM
PMF EMT is quadratic in the magnetic fields
therefore its Fourier transform is a convolution
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EXAMPLES OF THE RESULTS FOR THE EMT AND LORENTZ
FORCE CONVOLUTION
An analytical result valid for every generic
spectral index is that our spectrum goes to zero
for k2 kD.
Paoletti et al. 2008
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In all the figures the spectra are multiplies for
(n3)2 k3. The spectra are in units of
Scalar
Vector
Solid n-2.5 Longest dashed n3
Solid n-2.5 Longest dashed n3
Tensor spectra
Scalar Lorentz force
Solid n-2.5 Longest dashed n3
Solid n-2.5 Longest dashed n3
All the figures are taken from Paoletti et al.
2008
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RESULTS
All these theoretical results have been
implemented in the Einstein Boltzmann code CAMB
(httpcosmologist.info) where originally the
effects of PMF are considered only for vector
perturbations, anyway also this part of the code
has been improved by implementing the correct EMT
power spectrum. We implemented all the effects
mentioned above. In the following I am going to
show you some of the results of this
implementation.
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RESULTS FOR TEMPERATURE APS
Solid regula adiabatic mode Dotted scalar
mode Dashes tensor mode Dot-Dashesvector mode
N2
N-2.5
All the figures are taken from Paoletti et al.
2008
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RESULTS FOR TE CROSS-CORRELATION APS
Solid regula adiabatic mode Dotted scalar
mode Dashes tensor mode Dot-Dashesvector mode
N2
N-2.5
All the figures are taken from Paoletti et al.
2008
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RESULTS FOR E-MODE APS
Solid regula adiabatic mode Dotted scalar
mode Dashes tensor mode Dot-Dashesvector mode
N2
N-2.5
All the figures are taken from Paoletti et al.
2008
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RESULTS FOR B-MODE APS
Solid regula adiabatic mode Dotted
lensing Dashes tensor mode Dot-Dashesvector
mode
N2
N-2.5
All the figures are taken from Paoletti et al.
2008
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CONCLUSIONS
We have considered the effects of a SB of PMF on
the CMB anisotropies. We have considered the
magnetically induced perturbations of all kind
scalar vector and tensor . We calculated the
correct initial conditions for scalar and tensor
cosmological perturbations and showed the
behaviour of both the scalar and the tensor fully
magnetic mode and also the vector one sourced by
PMF. We calculated the exact PMF EMT power
spectra without any approximation. The results
show that the dominant contributions are the
scalar and the vector while the tensor one
remains subdominant. In particular the scalar
mode dominates on large scales while the vector
mode is the dominant contribution on small
scales. We showed that there are important
effects on CMB temperature and polarization APS.
Therefore present and future CMB data can
constrain PMF up to values for rms of few
nGauss.